Spectral Transformation Chains and Some New Biorthogonal Rational Functions

Abstract:A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation are revealed. A class of its rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed. The latter lead to large families of biorthogonal rational functions based upon the very-well-poised balanced hypergeometric series of three types: the standard hypergeometric series 9F8, basic series 10ϕ9 and its elliptic analogue 10E9. For an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly.

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